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The origin of the detrimental nonlinear response in high Tc superconductor (HTS) microwave devices is currently not well understood. In order to help elucidate the origin of these nonlinear effects, we have developed a description of the nonlinear response in superconductors in terms of a current-dependent complex conductivity. We demonstrate that such a treatment can consistently describe the results of power-dependent surface impedance measurements in resonator geometries as well as harmonic generation and intermodulation distortion effects in transmission line geometries. This approach yields a device-independent quantity that describes the nonlinear response of the superconducting material itself, which is suitable for comparisons of different materials and for material optimization. A further benefit of this description of nonlinear effects is that the relative importance of the nonlinear resistive and inductive components of a superconductor can be examined. We use this approach to predict the phase of the nonlinear response in HTS planar transmission lines, and compare our predictions with new phase-sensitive measurements made using a nonlinear vector network analyzer.